432edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br />8ve Stretch (¢) | ||
! colspan="2" | Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
| Line 46: | Line 47: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br />Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 60: | Line 62: | ||
|- | |- | ||
| 4 | | 4 | ||
| 179\432<br>(37\432) | | 179\432<br />(37\432) | ||
| 497.22<br>(102.78) | | 497.22<br />(102.78) | ||
| 4/3 | | 4/3 | ||
| [[Undim]] | | [[Undim]] | ||
|} | |} | ||
{{orf}} | |||
Revision as of 01:05, 16 November 2024
| ← 431edo | 432edo | 433edo → |
Theory
432edo has a reasonable approximation to the 7-limit, where the equal temperament tempers out 5120/5103, 703125/702464, 40353607/40310784, 102760448/102515625, and 283115520/282475249. It supports and provides a good tuning for the 5-limit maja temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.82 | -0.20 | +0.62 | -1.13 | -1.32 | +1.14 | +0.62 | +0.60 | -0.29 | -1.34 | -0.50 |
| Relative (%) | +29.6 | -7.3 | +22.3 | -40.8 | -47.4 | +41.0 | +22.3 | +21.6 | -10.5 | -48.1 | -17.9 | |
| Steps (reduced) |
685 (253) |
1003 (139) |
1213 (349) |
1369 (73) |
1494 (198) |
1599 (303) |
1688 (392) |
1766 (38) |
1835 (107) |
1897 (169) |
1954 (226) | |
Subsets and supersets
432 is a highly factorable number, factoring into 24 × 33, so 432edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [685 -432⟩ | [⟨432 685]] | -0.2596 | 0.2595 | 9.34 |
| 2.3.5 | [41 -20 -4⟩, [-3 -23 17⟩ | [⟨432 685 1003]] | -0.1440 | 0.2676 | 9.63 |
| 2.3.5.7 | 5120/5103, 703125/702464, 6565234375/6530347008 | [⟨432 685 1003 1213]] | -0.1631 | 0.2341 | 8.43 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 163\432 | 452.78 | 125/96 | Maja |
| 4 | 179\432 (37\432) |
497.22 (102.78) |
4/3 | Undim |